xrge0iifiso - Mathbox for Thierry Arnoux

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Theorem xrge0iifiso 31288

Description: The defined bijection from the closed unit interval onto the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)

**Hypothesis**
Ref	Expression
xrge0iifhmeo.1	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))

**Assertion**
Ref	Expression
xrge0iifiso	⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Distinct variable group: 𝑥,𝐹

Proof of Theorem xrge0iifiso Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 12808	. . 3 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 12522	. . 3 ⊢ < Or ℝ^*
3		soss 5457	. . 3 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. 2 ⊢ < Or (0[,]1)
5		iccssxr 12808	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
6		cnvso 6107	. . . . 5 ⊢ ( < Or ℝ^* ↔ ^◡ < Or ℝ^*)
7	2, 6	mpbi 233	. . . 4 ⊢ ^◡ < Or ℝ^*
8		sopo 5456	. . . 4 ⊢ (^◡ < Or ℝ^* → ^◡ < Po ℝ^*)
9	7, 8	ax-mp 5	. . 3 ⊢ ^◡ < Po ℝ^*
10		poss 5440	. . 3 ⊢ ((0[,]+∞) ⊆ ℝ^* → (^◡ < Po ℝ^* → ^◡ < Po (0[,]+∞)))
11	5, 9, 10	mp2 9	. 2 ⊢ ^◡ < Po (0[,]+∞)
12		xrge0iifhmeo.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))
13	12	xrge0iifcnv 31286	. . . 4 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑧 ∈ (0[,]+∞) ↦ if(𝑧 = +∞, 0, (exp‘-𝑧))))
14	13	simpli 487	. . 3 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
15		f1ofo 6597	. . 3 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)–onto→(0[,]+∞))
16	14, 15	ax-mp 5	. 2 ⊢ 𝐹:(0[,]1)–onto→(0[,]+∞)
17		0xr 10677	. . . . . . . 8 ⊢ 0 ∈ ℝ^*
18		1xr 10689	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
19		0le1 11152	. . . . . . . 8 ⊢ 0 ≤ 1
20		snunioc 12858	. . . . . . . 8 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
21	17, 18, 19, 20	mp3an 1458	. . . . . . 7 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
22	21	eleq2i 2881	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ 𝑤 ∈ (0[,]1))
23		elun 4076	. . . . . 6 ⊢ (𝑤 ∈ ({0} ∪ (0(,]1)) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
24	22, 23	bitr3i 280	. . . . 5 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)))
25		velsn 4541	. . . . . . 7 ⊢ (𝑤 ∈ {0} ↔ 𝑤 = 0)
26		elunitrn 12845	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
27	26	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ ℝ)
28		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 0 < 𝑧)
29		elicc01 12844	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
30	29	simp3bi 1144	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ≤ 1)
31	30	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ≤ 1)
32		1re 10630	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		elioc2 12788	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1)))
34	17, 32, 33	mp2an 691	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) ↔ (𝑧 ∈ ℝ ∧ 0 < 𝑧 ∧ 𝑧 ≤ 1))
35	27, 28, 31, 34	syl3anbrc 1340	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → 𝑧 ∈ (0(,]1))
36		pnfxr 10684	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
37		0le0 11726	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 0
38		ltpnf 12503	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℝ → 1 < +∞)
39	32, 38	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 1 < +∞
40		iocssioo 12817	. . . . . . . . . . . . . . 15 ⊢ (((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (0 ≤ 0 ∧ 1 < +∞)) → (0(,]1) ⊆ (0(,)+∞))
41	17, 36, 37, 39, 40	mp4an 692	. . . . . . . . . . . . . 14 ⊢ (0(,]1) ⊆ (0(,)+∞)
42		ioorp 12803	. . . . . . . . . . . . . 14 ⊢ (0(,)+∞) = ℝ⁺
43	41, 42	sseqtri 3951	. . . . . . . . . . . . 13 ⊢ (0(,]1) ⊆ ℝ⁺
44	43	sseli 3911	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0(,]1) → 𝑧 ∈ ℝ⁺)
45		relogcl 25167	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℝ⁺ → (log‘𝑧) ∈ ℝ)
46	45	renegcld 11056	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) ∈ ℝ)
47		ltpnf 12503	. . . . . . . . . . . . . 14 ⊢ (-(log‘𝑧) ∈ ℝ → -(log‘𝑧) < +∞)
48	46, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → -(log‘𝑧) < +∞)
49		brcnvg 5714	. . . . . . . . . . . . . 14 ⊢ ((+∞ ∈ ℝ^* ∧ -(log‘𝑧) ∈ ℝ) → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
50	36, 46, 49	sylancr 590	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℝ⁺ → (+∞^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < +∞))
51	48, 50	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → +∞^◡ < -(log‘𝑧))
52	44, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < -(log‘𝑧))
53	12	xrge0iifcv 31287	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0(,]1) → (𝐹‘𝑧) = -(log‘𝑧))
54	52, 53	breqtrrd 5058	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0(,]1) → +∞^◡ < (𝐹‘𝑧))
55	35, 54	syl 17	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 0 < 𝑧) → +∞^◡ < (𝐹‘𝑧))
56	55	ex 416	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,]1) → (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧)))
57		breq1 5033	. . . . . . . . 9 ⊢ (𝑤 = 0 → (𝑤 < 𝑧 ↔ 0 < 𝑧))
58		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = (𝐹‘0))
59		0elunit 12847	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
60		iftrue 4431	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → if(𝑥 = 0, +∞, -(log‘𝑥)) = +∞)
61		pnfex 10683	. . . . . . . . . . . . 13 ⊢ +∞ ∈ V
62	60, 12, 61	fvmpt 6745	. . . . . . . . . . . 12 ⊢ (0 ∈ (0[,]1) → (𝐹‘0) = +∞)
63	59, 62	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹‘0) = +∞
64	58, 63	eqtrdi 2849	. . . . . . . . . 10 ⊢ (𝑤 = 0 → (𝐹‘𝑤) = +∞)
65	64	breq1d 5040	. . . . . . . . 9 ⊢ (𝑤 = 0 → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ +∞^◡ < (𝐹‘𝑧)))
66	57, 65	imbi12d 348	. . . . . . . 8 ⊢ (𝑤 = 0 → ((𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)) ↔ (0 < 𝑧 → +∞^◡ < (𝐹‘𝑧))))
67	56, 66	syl5ibr 249	. . . . . . 7 ⊢ (𝑤 = 0 → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
68	25, 67	sylbi 220	. . . . . 6 ⊢ (𝑤 ∈ {0} → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
69		simpll 766	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ (0(,]1))
70	26	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ ℝ)
71		0re 10632	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
72	71	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 ∈ ℝ)
73	43	sseli 3911	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ⁺)
74	73	rpred 12419	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 𝑤 ∈ ℝ)
75	74	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 ∈ ℝ)
76		elioc2 12788	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1)))
77	17, 32, 76	mp2an 691	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0(,]1) ↔ (𝑤 ∈ ℝ ∧ 0 < 𝑤 ∧ 𝑤 ≤ 1))
78	77	simp2bi 1143	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0(,]1) → 0 < 𝑤)
79	78	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑤)
80		simpr 488	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑤 < 𝑧)
81	72, 75, 70, 79, 80	lttrd 10790	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 0 < 𝑧)
82	30	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ≤ 1)
83	70, 81, 82, 34	syl3anbrc 1340	. . . . . . . . 9 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → 𝑧 ∈ (0(,]1))
84	69, 83	jca 515	. . . . . . . 8 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)))
85	73	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑤 ∈ ℝ⁺)
86	85	relogcld 25214	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑤) ∈ ℝ)
87	44	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → 𝑧 ∈ ℝ⁺)
88	87	relogcld 25214	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (log‘𝑧) ∈ ℝ)
89	86, 88	ltnegd 11207	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((log‘𝑤) < (log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
90		logltb 25191	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
91	73, 44, 90	syl2an 598	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ (log‘𝑤) < (log‘𝑧)))
92		negex 10873	. . . . . . . . . . . . 13 ⊢ -(log‘𝑤) ∈ V
93		negex 10873	. . . . . . . . . . . . 13 ⊢ -(log‘𝑧) ∈ V
94	92, 93	brcnv 5717	. . . . . . . . . . . 12 ⊢ (-(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤))
95	94	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (-(log‘𝑤)^◡ < -(log‘𝑧) ↔ -(log‘𝑧) < -(log‘𝑤)))
96	89, 91, 95	3bitr4d 314	. . . . . . . . . 10 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
97	96	biimpd 232	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → -(log‘𝑤)^◡ < -(log‘𝑧)))
98	12	xrge0iifcv 31287	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0(,]1) → (𝐹‘𝑤) = -(log‘𝑤))
99	98, 53	breqan12d 5046	. . . . . . . . 9 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → ((𝐹‘𝑤)^◡ < (𝐹‘𝑧) ↔ -(log‘𝑤)^◡ < -(log‘𝑧)))
100	97, 99	sylibrd 262	. . . . . . . 8 ⊢ ((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0(,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
101	84, 80, 100	sylc 65	. . . . . . 7 ⊢ (((𝑤 ∈ (0(,]1) ∧ 𝑧 ∈ (0[,]1)) ∧ 𝑤 < 𝑧) → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
102	101	exp31 423	. . . . . 6 ⊢ (𝑤 ∈ (0(,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
103	68, 102	jaoi 854	. . . . 5 ⊢ ((𝑤 ∈ {0} ∨ 𝑤 ∈ (0(,]1)) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
104	24, 103	sylbi 220	. . . 4 ⊢ (𝑤 ∈ (0[,]1) → (𝑧 ∈ (0[,]1) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))))
105	104	imp 410	. . 3 ⊢ ((𝑤 ∈ (0[,]1) ∧ 𝑧 ∈ (0[,]1)) → (𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))
106	105	rgen2 3168	. 2 ⊢ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧))
107		soisoi 7060	. 2 ⊢ ((( < Or (0[,]1) ∧ ^◡ < Po (0[,]+∞)) ∧ (𝐹:(0[,]1)–onto→(0[,]+∞) ∧ ∀𝑤 ∈ (0[,]1)∀𝑧 ∈ (0[,]1)(𝑤 < 𝑧 → (𝐹‘𝑤)^◡ < (𝐹‘𝑧)))) → 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞)))
108	4, 11, 16, 106, 107	mp4an 692	1 ⊢ 𝐹 Isom < , ^◡ < ((0[,]1), (0[,]+∞))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∪ cun 3879 ⊆ wss 3881 ifcif 4425 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 Po wpo 5436 Or wor 5437 ^◡ccnv 5518 –onto→wfo 6322 –1-1-onto→wf1o 6323 ‘cfv 6324 Isom wiso 6325 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 +∞cpnf 10661 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 -cneg 10860 ℝ⁺crp 12377 (,)cioo 12726 (,]cioc 12727 [,]cicc 12729 expce 15407 logclog 25146

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148

This theorem is referenced by: xrge0iifhmeo 31289

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